SpheroidalS1Prime

SpheroidalS1Prime[n,m,γ,z]

gives the derivative with respect to of the radial spheroidal function of the first kind.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For certain special arguments, SpheroidalS1Prime automatically evaluates to exact values.
  • SpheroidalS1Prime can be evaluated to arbitrary numerical precision.
  • SpheroidalS1Prime automatically threads over lists.

Examples

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Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at a singular point:

Scope  (22)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

Specific Values  (4)

Simple exact values are generated automatically:

Find the first positive maximum of SpheroidalS1Prime[2,0,5,x]:

SpheroidalS1Prime functions become elementary if m=1 and γ=n π/2 :

TraditionalForm typesetting:

Visualization  (3)

Plot the SpheroidalS1Prime function for integer orders:

Plot the SpheroidalS1Prime function for noninteger parameters:

Plot the real part of TemplateBox[{2, 0, 1, {x, +, {i, , y}}}, SpheroidalS1Prime]:

Plot the imaginary part of TemplateBox[{2, 0, 1, {x, +, {i, , y}}}, SpheroidalS1Prime]:

Function Properties  (5)

SpheroidalS1Prime is not an analytic function:

TemplateBox[{1, 2, {pi, /, 2}, x}, SpheroidalS1Prime] has both singularities and discontinuities for :

TemplateBox[{1, 2, {pi, /, 2}, x}, SpheroidalS1Prime] is neither non-decreasing nor non-increasing:

TemplateBox[{1, 2, {pi, /, 2}, x}, SpheroidalS1Prime] is not injective:

SpheroidalS1Prime is neither non-negative nor non-positive:

SpheroidalS1Prime is neither convex nor concave:

Differentiation  (2)

First derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z when n=10, m=2 and γ=1/3:

Integration  (2)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

Series Expansions  (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

Applications  (1)

Find resonant frequencies for the Neumann problem in a prolate spheroidal cavity:

Determine the first few frequencies:

Possible Issues  (1)

Spheroidal functions do not evaluate for half-integer values of n and generic values of m:

Wolfram Research (2007), SpheroidalS1Prime, Wolfram Language function, https://reference.wolfram.com/language/ref/SpheroidalS1Prime.html.

Text

Wolfram Research (2007), SpheroidalS1Prime, Wolfram Language function, https://reference.wolfram.com/language/ref/SpheroidalS1Prime.html.

CMS

Wolfram Language. 2007. "SpheroidalS1Prime." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SpheroidalS1Prime.html.

APA

Wolfram Language. (2007). SpheroidalS1Prime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SpheroidalS1Prime.html

BibTeX

@misc{reference.wolfram_2023_spheroidals1prime, author="Wolfram Research", title="{SpheroidalS1Prime}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SpheroidalS1Prime.html}", note=[Accessed: 02-October-2023 ]}

BibLaTeX

@online{reference.wolfram_2023_spheroidals1prime, organization={Wolfram Research}, title={SpheroidalS1Prime}, year={2007}, url={https://reference.wolfram.com/language/ref/SpheroidalS1Prime.html}, note=[Accessed: 02-October-2023 ]}